Si函数

Si 函数定义如下

{\it Si} \left( z \right) =\int _{0}^{z}\!{\frac {\sin \left( t \right) }{t}}{dt}

Si(z)是下列三階常微分方程的一个解:

{\it Si} \left( z \right) =z{\frac {d}{dz}}w \left( z \right) +2\,{
\frac {d^{2}}{d{z}^{2}}}w \left( z \right) +z{\frac {d^{3}}{d{z}^{3}}}
w \left( z \right) =0

即：

w \left( z \right) ={\it \_C1}+{\it \_C2}\,{\it Si} \left( z \right)
+{\it \_C3}\,{\it Ci} \left( z \right)

表示为其他特殊函数
Meijer G-函数
*
*\frac{ -1}{2}\,\sqrt {\pi }
G^{1, 1}_{1, 3}\left(1/4\,{z}^{2}\, \Big\vert\,^{1}_{1/2, 0, 0}\right)
超几何函数

Si(z)=z_{1}F_{2}(1/2;3/2, 3/2; -(1/4)*z^2)

级数展开

• {\it Si} \left( z \right) =(z-{\frac {1}{18}}{z}^{3}+{\frac {1}{600}}{

z}^{5}-{\frac {1}{35280}}{z}^{7}+{\frac {1}{3265920}}{z}^{9}-{\frac {1
}{439084800}}{z}^{11}+{\frac {1}{80951270400}}{z}^{13}+O \left( {z}^{
15} \right) )

帕德近似
Si(z) \approx \left( -{\frac {33317056220720070437}{9686419676455776844590000}}\,{z
}^{7}+{\frac {67177799936189717}{98024149196718942600}}\,{z}^{5}-{
\frac {540705278447237}{16111793096107650}}\,{z}^{3}+z \right)
\left( 1+{\frac {177197169001594}{8055896548053825}}\,{z}^{2}+{\frac
{87368534024947}{363052404432292380}}\,{z}^{4}+{\frac {212787117226481
}{131788022808922133940}}\,{z}^{6}+{\frac {10065927082366801}{
1707972775603630855862400}}\,{z}^{8} \right) ^{-1}

图集
参见
*Sinhc 函数
*Coshc 函数
*Tanc 函数
*Tanhc 函数
*Chi 函数

参考文献